Thick film resistors are employed in hybrid electronic circuits to provide a wide range of resistor values, generally between about 0.1 .OMEGA. and about 10 .OMEGA.. Such resistors are printed on a ceramic substrate using thick film pastes, or inks, which are conventionally composed of an organic vehicle, a glass frit composition, an electrically conductive material, and various additives used to favorably effect the final electrical properties of the resistor. Theoretically, a single ink composition could be used to create all resistors on a given circuit by forming the resistors to have appropriate lengths. However, space and size constraints typically dictate the use of different inks compositions within a given circuit. For this purpose, inks are commercially available in composition families referred to as end-members, which are formulated to produce resistors having sheet resistivities (R.sub.S) in decade values from about 1 ohm per square (.OMEGA./.quadrature.) to about 10 megohms per square (M.OMEGA./.quadrature.), (per 25 micrometers of dried thickness). Compositions having values that are one decade apart are referred to as adjacent end-members, which are blended to produce intermediate values of resistance.
After printing, thick film inks are typically dried and then sintered, or fired, to convert the ink into a suitable film that adheres to the ceramic substrate. During sintering, the ink is heated at a rate that is sufficiently slow to promote stability of the resistor and to allow the organic vehicle of the ink to burnoff. Both physical and chemical changes occur within the thick film during sintering, by which the conduction network or microstructure of the resistor are formed. Various additives are typically used to achieve specific desired resistivity, stability and temperature characteristics.
The electrical resistance of a thick film resistor will vary with temperature, and may be permanently altered when subjected to a hostile environment. A thick film resistor's sensitivity to temperature is indicated by its temperature coefficient of resistance (TCR), as measured in parts per million per degree C. (ppm/.degree.C.). Thick film resistors can typically be calibrated to have a TCR in the range of about .+-.50 to about .+-.100 ppm/.degree.C. Calibration to a tighter limit is generally prevented by a significant difference in the values for TCR obtained at -55.degree. C. and 125.degree. C., which are standard temperature extremes used by the industry to evaluate the electrical characteristics of thick film resistors, as well as blending anomalies which occur as a result of interactions between the additives included in the ink to selectively alter the electrical characteristics of the resistor.
The resistance of a thick film resistor can be theoretically determined by the following equation: EQU Equation (1) Resistance (.OMEGA.)=R.sub.S .times.L/W
where R.sub.S is the sheet resistivity of the ink composition in ohms/square (.OMEGA./.quadrature.), L is the electrical length of the resistor, and W is the electrical width of the resistor. This relationship is conventionally used to design thick film resistors for hybrid circuits, with the length (L) of the resistor often being the final design characteristic manipulated to obtain the targeted resistance for a resistor in a circuit.
In practice, the behavior defined by Equation (1) above is non-ideal, with as-fired thick film resistors having lower resistances than that predicted by the ideal Equation (1). Generally, the sheet resistivity value of a resistor decreases as the length of the resistor decreases due to metal ion (conductor) diffusion into the resistor during firing, such as when silver-bearing thick film conductors are employed to terminate the resistor on the circuit. Changes in the TCR value of a resistor also occur, in that TCR values are a function of sheet resistance. The degree of conductor diffusion is relatively constant for a particular resistor ink-conductor ink combination. For very long resistors, the degree of diffusion may represent an insignificant portion of the resistor area, such that the effect on sheet resistivity may not be significant. However, for relatively short resistors, the same degree of diffusion represents a greater proportion of the resistor area, such that the effect of conductor diffusion on sheet resistivity can be significant, yielding an "out of balance" resistor whose resistance is below that required by its hybrid electronic circuit. Consequently, the above ideal Equation (1) cannot be used to accurately determine the resistance value of an as-fired thick film resistor, because the sheet resistance value of a given ink composition will change as a result of diffusion during firing.
As a result, thick film resistors must typically be trimmed to effectively increase their electrical length, and thereby increase their resistance values to that required by their circuits. While final resistance values of about .+-.1% can be achieved using abrasive or laser trimming techniques, the added processing step is undesirable from the standpoint of production costs and throughput, as well as reliability and stability of the resulting resistor. Generally, the degree to which the resistance value of a resistor can be corrected by trimming is limited by reliability considerations, such that values outside a specified range may result in its circuit being scrapped. Consequently, the ability to reduce or eliminate the requirement for trimming would enhance the reliability of the circuit and promote higher production rates.
Because trimming effectively increases the length of a resistor but does not change the sheet resistivity of the resistor composition, the TCR value of a resistor remains unchanged by the trimming process. Consequently, the TCR values of thick film resistors formed of the same ink can vary significantly from each other, particularly if the resistors have different aspect ratios (the length/width ratio of a resistor). Differences in TCR values between two or more resistors in a circuit are referred to as "TCR tracking." Many hybrid circuits require specific TCR tracking in order to perform appropriately under extreme thermal conditions. The degree of success in producing such circuits is therefore a function of the lengths of the resistors as a result of the tendency for conductor diffusion and its effect on the sheet resistivity and TCR value of a resistor.
In view of the above complications, current methodologies employed in the prior art to design thick film resistors include creating designs based on the ideal Equation (1), and then employing trial and error iterations to balance the resistors relative to the resistance values and TCR tracking required by a circuit. However, such an approach may take many iterations that can span several years. This is due largely to the nature of the trial and error balancing method, which does not enable any apparent imbalance to be identified as one that is specifically driven by the non-ideal behavior of the Equation (1) relationships or by variables of the printing and firing processes. Consequently, design iterations in which the dimensions of a resistor are adjusted in order to achieve a required resistor balance and/or TCR tracking are made unnecessarily if the true culprit is printer setup or temperature uniformity within the sintering furnace. As a result, as subsequent circuits are produced, slight differences in printer setup and/or firing parameters may necessitate yet another iteration to re-attain the required resistor balance and/or TCR tracking.
Another technique that can be used in conjunction with the iterate method described above is to reduce the degree of conductor diffusion into the resistor during sintering. Such a technique may involve the adding of diffusion blockers to the resistor ink composition, and/or employing thick film conductor inks that exhibit a low diffusion potential relative to the thick film resistor material. As such, this technique is intended to minimize the effect that conductor diffusion has on the resistivity and TCR value of a resistor. While such a solution may lessen the otherwise intense iterative method described above, current production ink compositions have not been effective enough to eliminate the requirement for post-firing trimming or achieve a desired level of TCR tracking.
From the above, it can be seen that present practices involving the processing of thick film resistors are generally inexact in terms of producing resistors which can be accurately and repeatably processed to exhibit resistance values and TCR tracking required by their hybrid electronic circuits. In particular, present practices generally necessitate numerous design iterations and time-consuming in-process trimming operations in order to attain the resistance and TCR values required by a circuit. Furthermore, prior art methods do not enable resistance values and TCR tracking targets to be readily achieved by thick film resistors in their as-fired condition. Accordingly, what is needed is a method for producing thick film resistors, in which the dimensions of an as-fired resistor can be accurately specified in the design stage so as to more readily achieve resistance values and balance between resistors of a circuit, even where such resistors have significantly different aspect ratios. It would also be desirable that such a method enable the production of resistors from a single ink to have near-constant TCR values, regardless of the physical sizes of the resistors, so as to improve TCR tracking.